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Chapter 8: Problem 19

A polyhedron has 12 vertices and 30 edges. How many faces does it have?

Short Answer

Expert verified
The polyhedron has 20 faces.

Step by step solution

01

Understand Euler's Formula

Euler's Formula for polyhedra states that for any convex polyhedron, the relationship between vertices (V), edges (E), and faces (F) is given by: \[ V - E + F = 2 \]
02

Substitute the Given Values

Substitute the given values from the exercise into Euler's Formula. Here, we have 12 vertices (V=12) and 30 edges (E=30). Therefore, the equation becomes: \[ 12 - 30 + F = 2 \]
03

Solve for the Number of Faces

Rearrange the equation to solve for the number of faces (F). \[ 12 - 30 + F = 2 \] Simplify to find: \[ F = 2 + 30 - 12 \] \[ F = 20 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertices
In the context of polyhedra, vertices are the points where two or more edges meet. Think of each vertex as a corner of the shape. For example, in a cube, each of the eight corners is a vertex, making a total of 8 vertices.

In the given exercise, the polyhedron has 12 vertices. These are the key points where the edges (which are the lines connecting vertices) converge. Identifying the vertices is the first step in understanding the structure of a polyhedron. Recognizing this helps make it easier to apply Euler's Formula later.
Edges
Edges are the line segments that connect two vertices in a polyhedron. If you imagine a cube again, each side of a square that makes up the cube is an edge. In total, a cube has 12 edges.

For the polyhedron in the exercise, it has 30 edges. When we look at Euler's Formula \( V - E + F = 2 \), edges are represented by \( E \).

In this scenario, we know \( E = 30 \).

The edges serve as a crucial linking structure for vertices and faces, helping to shape the three-dimensional form of the polyhedron.
Faces
Faces are the flat surfaces enclosed by edges on a polyhedron. Imagine each square on a cube - there are six such squares, and each one is a face. Faces are bounded by edges and typically meet at vertices.

In the exercise, we are asked to calculate the number of faces using Euler's Formula: \( V - E + F = 2 \). We substitute the given values:
We know the formula will look like this: \( 12 - 30 + F = 2 \)
Simplifying it further, we have: \( F = 20 \).

Therefore, the polyhedron in question has 20 faces. Faces contribute to the overall shape and form of the polyhedron, and understanding how to count them is fundamental in the study of geometrical shapes.

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